Let$f$and$g$be nonlinear entire functions. The relations between the dynamics of$f⊗g$and$g⊗f$are discussed. Denote byℐ (·) and$F$(·) the Julia and Fatou sets. It is proved that if$z$∈$C$, then$z$∈ℐ8464 ($f⊗g$) if and only if$g(z)$∈ℐ8464 ($g⊗f$); if$U$is a component of$F$($f$○$g$) and$V$is the component of$F$($g$○$g$) that contains$g(U)$, then$U$is wandering if and only if$V$is wandering; if$U$is periodic, then so is$V$and moreover,$V$is of the same type according to the classification of periodic components as$U$. These results are used to show that certain new classes of entire functions do not have wandering domains.

This paper studies Banach space valued Hausdorff-Young inequalities. The largest part considers ways of changing the underlying group. In particular the possibility to deduce the inequality for open subgroups as well as for quotient groups arising from compact subgroups is secured. A large body of results concerns the classical groups$T$^{$n$},$R$^{$n$}and$Z$_{$k$}. Notions of Fourier type are introduced and they are shown to be equivalent to properties expressed by finite groups alone.

One of our main results is the following: Let$X$be a compact connected subset of the Euclidean space$R$^{$n$}and$r(X, d$_{2}) the rendezvous number of$X$, where$d$_{2}denotes the Euclidean distance in$R$^{$n$}. (The rendezvous number$r(X, d$_{2}) is the unique positive real number with the property that for each positive integer$n$and for all (not necessarily distinct)$x$_{1},$x$_{2},...,$x$_{$n$}in$X$, there exists some$x$in$X$such that $$(1/n)\sum\nolimits_{i = 1}^n {d_2 (x_i ,x)} = r(X,d_2 )$$ .) Then there exists some regular Borel probability measure μ_{0}on$X$such that the value of ∫_{$X$}$d$_{2}($x, y$)$d$μ_{0}($y$) is independent of the choice$x$in$X$, if and only if$r$($X, d$_{2}) = sup_{μ}∫_{$X$}∫_{$X$}$d$_{2}($x, y$)$d$μ($x$)$d$μ($y$), where the supremum is taken over all regular Borel probability measures μ on$X$.

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